An optimal design approach for tracking problems and its assessment against classical controllers

A mathematical model for the steering control of an automobile is described. The structure of the model derives from linear optimal discrete time preview control theory but it is non-linear. Its parameter values are obtained by heuristic methods, using insight gained from the linear optimal control theory. The driver model is joined to a vehicle dynamics model and the path tracking performance is demonstrated, using moderate manoeuvring and racing speeds. The model is shown to be capable of excellent path following and to be robust against changes in the vehicle dynamics. Application to the simulation of manoeuvres specified by an ideal vehicle path and further development of the model to formalise the derivation of its parameter values and to put it to other uses are discussed.

This paper deals with a theoretical study of the active control of purely axial 1D instabilities in multi-stage axial-flow compressors. The paper briefly considers the development of a suitable surge model and continues with the derivation of a controller using linear optimal control theory. The controller is specifically designed to suppress the instabilities predicted by the linearised form of the surge model. The control technique involves a bleed being dynamically varied in response to fluctuations of variables. It is found that a stabilising optimal controller can always be obtained. The second part of the paper presents a non-linear simulation of the surge prediction model with the linear controller, using a 4th order Runge-Kutta scheme. This demonstrates how some non-linearities in the flow model can affect the controller operation. It is also shown that, the amplitudes and the frequency response required of the actuator to retain stability can exceed the practical limitations.Copyright © 1993 by ASME

Linear optimal control theory is applied to the control synthesis of a high bypass ratio, variable pitch, turbofan engine. The basic control philosophy is to use only a low order dynamic model of the plant coupled with the concept of integral-output states so as to maintain control simplicity yet guarantee integral control of thrust, turbine temperature and other important engine outputs. Linear simulation results indicate that the control system developed provides rapid control of small thrust perturbations and quickly eliminates the effect of unmodelled thrust and temperature disturbances. Large thrust accelerations are obtained in about one half second while the control maintains negligible overshoot in temperature and stall margins.

The development of on-ramp metering control strategies for a freeway can be aided by the well-developed theory of linear optimal control. However, an adequate model of an appropriate form must be available. Recently developed "aggregate variable" models appear to be best suited to this purpose. In these models the freeway is sectioned into lengths of about <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\frac{1}{2}</tex> mi and traffic is described in terms of the aggregate variables section speed, section density, and volume. The model is used to formulate a linear regulator problem with quadratic cost for the purpose of determining a feedback control rule which returns the freeway to nominal conditions after a disturbance. The solution of the regulator problem involves considerable computation difficulty when the order of the system is large. In a large class of applications the optimal control law has a local structure which can be exploited in a computationally efficient manner to yield a suboptimal control with little loss of system performance. A new technique, the augmentation method, is presented for constructing a suboptimal control from a set of optimal controls for low-order sub-systems. New results are presented which provide a rational and elegant measure for the additional cost associated with an arbitrary stabilizing suboptimal control. Additional results pertain especially to the augmentation procedure. The models and suboptimal control scheme are applied to a specific detailed design for an 8-mi segment of the San Diego Freeway in Los Angeles. Simulations are shown illustrating the effectiveness of such traffic-responsive controls in clearing up congestion due to traffic incidents.

The development of on ramp metering control strategies for a freeway can be aided by the well-developed theory of linear optimal control. The recently developed variable model appears to be best suited for this purpose. In this model the freeway is partitioned into lengths of about 1/2 mile and traffic is described in terms of the aggregate variables section speed, section density, and volume. This model and linear optimal control theory are used to developed an on ramp metering strategy which is designed to maintain freeway conditions near nominal conditions in the presence of disturbances. An illustrative design for a segment of the San Diego freeway in Los Angeles is provided. The companion problem of processing freeway sensor data to provide current estimates of the aggregate variables can be approached through standard techniques in estimation theory. Methods of attacking this portion of the freeway surveillance problem are presented.

The use of an adaptive optimal control algorithm for two real-time control applications, an optimal excitation control of a synchronous generator (OEC) and a power system stabilizer (PSS), is described. Experimental studies on a physical model of a power system showed that the proposed OEC and PSS can track the controlled system by parameter identification at different operating conditions. The proposed algorithm is based on linear optimal control theory. A special fifth-order discrete Riccati equation is solved in each sampling period. The proposed OEC and PSS can always guarantee that in a closed loop the controlled system will be stable based on the identified parameters. As the actual output of the controlled system and control are directly used in the controller, no observer is required. The proposed OEC and PSS can track the controlled system very rapidly. A number of tests have been performed, and all show satisfactory results.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

A stability and control augmentation system (SCAS) was designed based on a set of comprehensive performance criteria. Linear optimal control theory was applied to determine appropriate feedback gains for the stability augmentation system (SAS). The helicopter was represented by six-degree-of-freedom rigid body equations of motion and constant factors were used as weightings for state and control variables. The ratio of these factors was employed as a parameter for SAS analysis and values of the feedback gains were selected on this basis to satisfy three of the performance criteria for full and partial state feedback systems. A least squares design method was then applied to determine control augmentation system (CAS) cross feed gains to satisfy the remaining seven performance criteria. The SCAS gains were then evaluated by nine degree-of-freedom equations which include flapping motion and conclusions drawn concerning the necessity of including the pitch/regressing and roll/regressing modes in SCAS analyses.

A fuzzy logic adaptive Kalman filtering methodology was developed for the automatic control of an irrigation canal system under unknown disturbances (water withdrawals) acting in the canal. Using a linearized finite difference model of open channel flow, the canal operation problem was formulated as an optimal control problem and an algorithm for gate opening in the presence of arbitrary external disturbances (changes in flow rates) was derived. Based on the linear optimal control theory, the linear quadratic regulator (LQR), assuming all the state variables (flow depths and flow rates) were available, was designed to generate control input (optimal gate opening). As it was expensive to measure all the state variables (flow rates and flow depths) in a canal system, a fuzzy logic adaptive Kalman filter and traditional Kalman filter were designed to estimate the values for the state variables that were not measured but were needed in the feedback loop. The performances of the state estimators designed using the fuzzy logic adaptive Kalman filter methodology and the traditional Kalman filtering technique were compared with the results obtained using the LQR (target loop function). The results of the present study indicated that the performance of the fuzzy logic adaptive Kalman filter was far superior to the performance of the observer design based upon the traditional Kalman filter approach. The obvious advantages of the fuzzy logic adaptive Kalman filter were the prevention of filter divergence and ease of implementation. As the fuzzy logic adaptive Kalman filter requires smaller number of state variables for the acceptable accuracy therefore, it would need less computational effort in the control of irrigation canals. Copyright © 2009 John Wiley &amp; Sons, Ltd.

A recurrent dynamic neural network (RDNN) based on Hopfield model is applied to linear quadratic regulator controller of an irrigation canal. The Saint-Venant equations of open-channel flow are linearized using the Taylor series and a finite difference approximation of the original nonlinear, partial differential equations. Using the linear optimal control theory, a Linear Quadratic Regulator (LQR) controller is developed for an irrigation canal with a single pool. A Hopfield recurrent dynamic neural network is applied to solution of Riccati equation for calculating the LQR control gains. The main advantage of Hopfield neural network model is its robustness and flexibility when dealing with unknown perturbations and parameters. The results of this study shows that an RDNN based LQR controller (RDNN/LQR) for irrigation canals offers an alternative to traditional LQR controller when dealing with the disturbances in the irrigation canals.

Combined Feedback Control of Metro Line and Train Operation

A EROSPACE engineering applications greatly stimulated the development of optimal control theory during the 1950s and 1960s, where the objective was to drive the system states in such a way that some defined cost was minimized. This turned out to have very useful applications in the design of regulators (where some steady state is to be maintained) and in tracking control strategies (where some predetermined state trajectory is to be followed). Among such applications was the problem of optimal flight trajectories for aircraft and space vehicles. Linear optimal control theory in particular has been very well documented and widely applied, where the plant that is controlled is assumed linear and the feedback controller is constrained to be linear with respect to its input. However, the availability of powerful low-cost microprocessors has spurred great advantages in the theory and applications of nonlinear control. The competitive era of rapid technological change, particularly in aerospace exploration, now demands stringent accuracy and cost requirements in nonlinear control systems. This has motivated the rapid development of nonlinear control theory for application to challenging, complex, dynamical real-world problems, particularly those that bear major practical significance in aerospace, marine, and defense industries. Infinite-time horizon nonlinear optimal control (ITHNOC) presents a viable option for synthesizing stabilizing controllers for nonlinear systems by making a state-input tradeoff, where the objective is to minimize the cost given by a performance index. The original theory of nonlinear optimal control dates from the 1960s. Various theoretical and practical aspects of the problem have been addressed in the literature over the decades since. In particular, the continuous-time nonlinear deterministic optimal control problem associated with autonomous (time-invariant) nonlinear regulator systems that are affine (linear) in the controls has been studied by many authors. The long-established theory of optimal control offers quite mature and well-documented techniques for solving this control-affine nonlinear optimization problem, based on dynamic programming or calculus of variations, but their application is generally a very tedious task. Bellman’s dynamic programming approach reduces to solving a nonlinear first-order partial differential equation (PDE), expressed by the Hamilton–Jacobi–Bellman (HJB) equation. The solution to the HJB equation gives the optimal performance/cost value (or storage) function and determines an optimal control in feedback form under some smoothness assumptions. Alternatively, in the classical calculus of variations, optimal control problems can be characterized locally in terms of the Hamiltonian dynamics arising from Pontryagin’s minimum principle. These are the characteristic equations of the HJB PDE, which result in a nonlinear, constrained two-point boundary value problem (TPBVP) that, in general, can only be solved by successive approximation of the optimal control input using iterative numerical techniques for each set of initial conditions. Numerically, even though the nonlinear TPBVP is somewhat easier to solve than the HJBPDE, control signals can only be determined offline and are thus best suited for feedforward control of plants for which the state trajectories are known a priori. Therefore, contrary to the dynamic programming approach, the resultant control law is not generally in feedback form. Open-loop control, however, is sensitive to random disturbances and requires that the initial state be on the optimal trajectory. In contrast, nonlinear optimal feedback has inherent robustness properties (inherent in the sense that it is obtained by ignoring uncertainty and disturbances). The potential difficulty with the HJB approach is that no efficient algorithm is available to solve the PDE when it is nonlinear and the problem dimension is high, making it impossible to derive exact expressions for optimal controls for most nontrivial problems of interest. The optimal can only be computed in special cases, such as linear dynamics and quadratic cost, or very low-dimensional systems. In particular, if the plant is linear time invariant (LTI) and the (infinite-time) performance index is quadratic, then the corresponding HJB equation for this infamous linear-quadratic regulator (LQR) problem reduces to an algebraic Riccati equation (ARE). Contrary to the well-developed and widely applied theory and computational tools for theRiccati equation (for example, see [1]), theHJB equation is difficult, if not impossible, to solve for most practical applications. The exact solution for the optimal control policies is very complex

A new optimal coordinated control strategy between an electronic power transformer (EPT) and generator excitation is proposed to improve power system dynamic performance and stability. The studied system is based on a single-machine multidouble-circuit transmission-lines (SMTL) system. In the SMTL system, the generator is connected to an infinite bus through a conventional power transformer with a double-circuit transmission line, and to another infinite bus with an EPT with a double-circuit transmission line. The mathematical model of the SMTL system is established, and the proposed optimal coordinated control strategy is upon linear optimal control theory. Numerical simulations were carried out for the SMTL system, and the simulation results showed that the proposed control strategy has good dynamical performance for a variety of disturbances, and can improve the system damping and voltage performance effectively.

This paper examines the problem of designing a linear feedback system so that its response to a specified input is relatively insensitive to slow changes in system parameters. Classical feedback design techniques involve the specification of the system sensitivity function on the basis only of the forced response to a given input. A new performance criterion has been derived, in which the mean square variation of the system response is minimized. This specification of the sensitivity function results in control over the variation of both the characteristic and forced responses with system changes. The approach is based upon the assumption of a differential change in the variable system parameter, but yields workable results for large changes. It employs mathematical techniques which have been well developed for other applications in network design. The stability problem associated with high loop gains is considerably reduced when the sensitivity function is specified on the basis of this criterion.

A method for the design and synthesis of near-optimal, nonlinear control laws is examined, based on a generalisation of linear quadratic optimal control theory and which effectively provides a near-optimal gain schedule. The method is simple to apply and affords greater design flexibility (via state-dependent weighting) than conventional approaches. The resulting regulator can, in principle, be implemented in real time owing to the causal nature of the required computations. However, the need to solve an algebraic Riccati equation at every time point is burdensome, and a number of algorithms that would permit parallel computation are discussed. The problem of stabilising an inverted pendulum is used to illustrate the method and proves an exacting task that highlights a number of issues.

Optimal guidance of extended trajectory shaping